State the formula for the Laplace transform of a function $f(t)$, defined for $t \geqslant 0$.

Let $f(t)$ be periodic with period $T$ (i.e. $f(t+T)=f(t)$ ). If $g(t)$ is defined to be equal to $f(t)$ in $[0, T]$ and zero elsewhere and its Laplace transform is $G(s)$, show that the Laplace transform of $f(t)$ is given by

$$F(s)=\frac{G(s)}{1-e^{-s T}}$$

Hence, or otherwise, find the inverse Laplace transform of

$$F(s)=\frac{1}{s} \frac{1-e^{-s T / 2}}{1-e^{-s T}}$$